1. Binary systems as sources of gravitational waves
Theoretical research done at Nikhef/Vu
Gideon Koekoek March 5th 2008

2. Overview of presentation
Introduction: the binary system
The Martel-Poisson method
Methods used: current research
Conclusions & outlook

3. Introduction
General Relativity: any system with a quadrupole moment produces GWs
Our prime target: the binary system!
A sure source of GW’s: Hulse & Taylor
Their GWs are about to be measured (LIGO, VIRGO experiments)
Test-bed for GR and extensions

4. Introduction How to calculate gravitational waves? General scheme:
Curvature of spacetime
Source of the GWs
General scheme:
Take background metric gBμν add a disturbance hμν and plug into the Einstein Equations
Differential equations that couple the gravitational wave hμν to the energy momentum Tμν

5. Introduction This is all well-known…
In the case of a Minkowski background, this is an easy exercise.
All constants
The wave equation!
Waves move with speed of light
Waves have two (!) polarisations
Are transversal
This is all well-known…

6. M Introduction m How about the binary system? Variables!
Approximation: one of the stars is very much heavier than the other (EMRI). The metric of the system is then that of the heavy star: Schwarzschild metric.
Variables!

7. Introduction The challenge: Difficult! How to solve this system?
Also, the source Tμν will be difficult, as the stars orbit each other in a non-trivial way.
(Epicycle method; see last year’s presentation)
The challenge:
Non-constant background metric
Non-trivial energy-momentum tensor
10 coupled, partial differential equations in 10 variables
Difficult! How to solve this system?

8. Introduction Our goal:
Expert groups are working on this by numerical methods:
..but this is a very demanding computation!
Our goal:
Try to find the gravitational waves for the binary system in an analytical way. We do this by using a formalism developed recently by Martel & Poisson, and work our way from there.
(Frans Pretorius, 2006)

9. The Martel-Poisson method

10. The Martel-Poisson method
Following a program started by John Wheeler (1957), Martel & Poisson devised a formalism (2004)
The Martel-Poisson method
Linearize the Einstein Equations in hμν
Decompose the waves into tensorial spherical harmonics..
Plug into Einstein Equations and find the EOMs for the coefficients..
Do some very clever gauging to eliminate all unphysical degrees of freedom..
Think very hard..

11. The Martel-Poisson method
..in the end: only two linear, uncoupled Klein-Gordon equations remain.
Great simplification!
10 coupled, partial differential equations in 10 variables
The two Ψ(t,r) ’s (roughly) correspond to the two polarisations of the GWs. From these, we can directly calculate the GWs and the emitted energy!

12. The Martel-Poisson method
A closer look at these two differential equations
In which:
Da is the generally covariant derivative, determined by the Schwarzschild metric
VZM(r) and VRW(r) are ‘potentials’, fully determined by the Schwarzschild metric
SZM(t,r) and SRW(t,r) are sources, fully determined by the motion of the small star around the bigger one
These two equations can now be solved!

13. Methods used: current research

14. Methods used: current research
How to solve these two equations?
As always in such matters, there are two options:
Numerical integration: write a clever C++ program that takes the initial conditions and extrapolates from there
Approximation techniques: throw out some terms, do integral transforms, and try to find an approximate analytical solution
We are doing them both!

15. Methods used: current research
Analytical scheme: 4M
> 6M
Includes some approximations and is based on Laplace transforms.
Solutions found are integrals in which the orbit of the smaller star can be freely specified, i.e. solutions work for general orbits
status: implementing initial conditions; results expected shortly.
Work in progress; results expected soon
Numerical scheme:
Spacetime is divided up into grid cells, worldline of the smaller star is plotted
C++ code integrates the EOMs over the worldline
Status: a first version of the code is available, and is ondergoing testing.

16. Conclusions & outlook

17. Conclusions & outlook
Immediate future (i.e. working on this right now)
Solve Martel & Poisson’s two EOMs both numerically and via an analytical scheme; compare the results.
The greater plan:
Blend these two formalisms, so the EMRI-condition can be relaxed..even for alternative gravities!
Martel & Poisson’s method only works when one of the masses is small, because otherwise the curvature deviates from Schwarzschild
We know a formalism to calculate deviations of the metric

18. Conclusions & outlook Work in progress..but results expected soon!
Problem: the gravitational waves in a binary system are very challenging to calculate
EMRI approximation: by assuming one of the stars to be much heavier than the other, the background metric can be taken as Schwarzschild
Martel & Poisson method: enables us to find the GWs in a Schwarzschild spacetime for any source, by solving two scalar EOMs
Status: Solving the two EOMs, by an analytical scheme using Laplace Transforms; making progress but no results yet. A numerical code is being developed: results will be compared.
Work in progress..but results expected soon!

19. Any questions? gkoekoek@nikhef.nl